Chapter 1.5, Problem 34E

To determine

To solve: The equation $e^{0.05t}=3$ and support the solution graphically.

Answer to Problem 34E

The solution of the equation $e^{0.05t}=3$ is approximately $21.97$.

Explanation of Solution

Given information: The equation is $e^{0.05t}=3$.

Calculation:

Take the $\ln$ on both sides of the equation $e^{0.05t}=3$ and simplify it by Power Rule.

  $\begin{array}{c}\ln e^{0.05t}=\ln 3\\ \left(0.05t\right)\ln e=\ln 3\\ t=\frac{\ln 3}{0.05}\\ \approx 21.97\end{array}$

To verify the solution graphically, plot the graph of the equation $e^{0.05t}=3$ . The intersection point of these two graphs will be the solution.

Follow the steps to solve the equation by calculator:

• First press “ON” button on graphical calculator and press $\boxed{\text{Y}=}$ key.
• Enter right hand side of the equation after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2\text{nd}}\boxed{÷}\boxed{^}\boxed{0.05}\boxed{\text{X}}$ . Now, press the key $\boxed{\text{ENTER}}$ .
• Enter right left side of the equation after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{3}$ .
• Press $\boxed{\text{WINDOW}}$ key and adjust the window at $\left[-5,35\right]$ by $\left[-1,4\right]$ .
• Now, press keystrokes given below to find the value of $t$ as:

  $\boxed{2\text{nd}}\to \boxed{\text{TRACE}}\to \boxed{5}\to \boxed{\text{ENTER}}\to \boxed{\text{ENTER}}\to \boxed{\text{ENTER}}$

  

From the graph, it can be observed that the intersection point is $\left(21.97,3\right)$ at $y=3$.

Therefore, the solution of the equation $e^{0.05t}=3$ is approximately $21.97$.